Methyl Substitution Destabilizes Alkyl Radicals

Abstract We have quantum chemically investigated how methyl substituents affect the stability of alkyl radicals MemH3−mC⋅ and the corresponding MemH3−mC−X bonds (X = H, CH3, OH; m = 0 – 3) using density functional theory at M06‐2X/TZ2P. The state‐of‐the‐art in physical organic chemistry is that alkyl radicals are stabilized upon an increase in their degree of substitution from methyl<primary<secondary<tertiary, and that this is the underlying cause for the decrease in C−H bond strength along this series. Here, we provide evidence that falsifies this model and show that, on the contrary, the MemH3−mC⋅ radical is destabilized with increasing substitution. The reason that the corresponding C−H bond nevertheless becomes weaker is that substitution destabilizes the sterically more congested MemH3−mC−H molecule even more.


Computational details
All calculations were performed with the Amsterdam Density Functional (ADF) program unless otherwise stated. [1,2] Molecular orbitals (MOs) were expanded using a large uncontracted set of Slater-type orbitals (STO): TZ2P. [3] The TZ2P basis set is of triple-z quality, augmented by two sets of polarization functions. All electrons were treated variationally. The meta-hybrid generalized gradient approximation (GGA) functional M06-2X was used for calculating the geometries and energies. [4] M06-2X was chosen based on earlier works in which the performance of density functional methods was investigated on trends in R-X bond dissociation energies with R = Me, Et, i-Pr, t-Bu and various X (for instance H, CH3, Cl, and OH). [5,6] In addition, our computed values with M06-2X nicely recover experimental R-X BDE values (see Table S1 and relevant references therein). Our conclusion from the analyses at M06-2X/TZ2P, that both the radical and parent molecule are destabilized upon methyl substitution, is nicely reproduced when carried out with nine other functionals (in combination with the same TZ2P basis set), namely: a) BLYP, b) BP86, c) PBE, d) B3LYP, e) PBE0, f) ωB97, g) B2PLYP, h) B2TPLYP and i) rev-DSD-BLYP (see Table S2 and Figure S1). [7] No geometry restrictions were used unless otherwise stated. The radical fragments were treated spin-unrestricted and the PyFrag2019 program was used for analyzing the bond dissociation as a function of the MemH3-mC-X distance. [8] For ease, and only in the case of displaying overlaps, fragments without spin polarization are used. NBO analyses were performed with the Gaussian 09 rev. D01 program at M06-2X/cc-pVTZ level of theory on the geometries optimized at M06-2X/TZ2P in ADF to highlight the hyperconjugation interaction in the parent molecule and the radical (see Figures S10-S11). [9,10] Thermochemistry Enthalpies at 298.15 K and 1 atmosphere (∆H298) were calculated from electronic bond energies (∆E) and vibrational frequencies using standard thermochemistry relations for an ideal gas, according to Equation (2): [11] ΔH298 = ∆E + ΔEtrans,298 + ΔErot,298 + ΔEvib,0 + Δ(ΔEvib,0)298 + Δ(pV) Here, ΔEtrans,298, ΔErot,298 and ΔEvib,0 are the differences between the reactant and products in translational, rotational and zero-point vibrational energy, respectively. Δ(ΔEvib,0)298 is the change in the vibrational energy difference as one goes from 0 to 298.15 K. The vibrational energy corrections are identical to our frequency calculations. The molar work term Δ(pV) is (Δn)RT; Δn = +1 for one reactant dissociating into the two products. Thermal corrections for the electronic energy are neglected.

Activation strain and energy decomposition analysis
For the activation strain analysis (ASA), the bond energy ∆E [which also features in Eq.
(2)] between two fragments is made up of two major components: [12] ∆E = ∆Estrain + ∆Eint Here, the strain energy ∆Estrain is the amount of energy required to deform the fragments from their equilibrium structure to the geometry that they acquire in the overall complex. The interaction energy ∆Eint corresponds to the actual energy change when the geometrically deformed fragments are combined to form the overall complex.
We further analyze the interaction ∆Eint in the framework of the canonical Kohn-Sham molecular orbital (MO) model, by dissecting it through our canonical energy decomposition analyses (canonical EDA) into the electrostatic attraction, the Pauli repulsion and the (attractive) orbital interactions: [1,12] ∆Eint = ∆Velstat + ∆EPauli + ∆Eoi (4) The term ∆Velstat corresponds to the classical electrostatic interaction between the unperturbed charge distributions of the fragments in the geometry they possess in the complex. This term is usually attractive. The Pauli-repulsion ∆EPauli between these fragments comprises the destabilizing interactions, associated with the Pauli-principle for fermions, between occupied orbitals and is responsible for the steric repulsion. The orbital interaction ∆Eoi between these fragments in any MO model, and therefore also in Kohn-Sham theory, accounts for electron-pair bonding (the SOMO-SOMO interaction), charge transfer (empty/occupied orbital mixing between different fragments) and polarization (empty/occupied orbital mixing on one fragment due to the presence of another fragment). The orbital interaction energy ∆Eoi can be further decomposed into the contributions from each irreducible representation Γ of the interacting system. The use of M06-2X gives a term that cannot be decomposed, which is a correction term, such that the total orbital interaction is the correct one.

Voronoi Deformation Density (VDD) Charge
The electron density distribution is analyzed by using the Voronoi deformation density (VDD) method for atomic charges. [13] The VDD atomic charge Q A VDD is computed as the (numerical) integral of the deformation density Δρ(r) = ρ(r) -∑B ρB(r) in the volume of the Voronoi cell of atom A [Eq. (7)]. [14] The Voronoi cell of atom A is defined as the compartment of space bound by the bond midplanes on and perpendicular to all bond axes between nucleus A and its neighboring nuclei (cf. the Wigner-Seitz cells in crystals). [13] In Eq. (5), ρ(r) is the electron density of the molecule and ∑ ρ B (r) B the superposition of atomic densities ρB of a fictitious promolecule without chemical interactions that is associated with the situation in which all atoms are neutral. The interpretation of the VDD charge Q A VDD is rather straightforward and transparent. Instead of measuring the amount of charge associated with a particular atom A, Q A VDD directly monitors how much charge flows, due to chemical interactions, out of (Q A VDD > 0) or into (Q A VDD < 0) the Voronoi cell of atom A, that is, the region of space that is closer to nucleus A than to any other nucleus.               [a] Computed at M06-2X/TZ2P and, for each m, at equal substituent-carbon distances based on the geometry of MemH3-mC-H. For H3-C • : D3h irreps merge to C3v ones as: a1' + a2" = a1, a2' + a1" = a2 and e' + e" = e. With ∆Eoi correction term for hybrid functional.